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ON EUCLID S FIVE POSTULATES - Revista Brasileira de História
... triangle for every triple of straight lines. Thus, the explicit Euclidean definition alone cannot be transformed into a postulate. We need to reveal a not explicitly mentioned characterization of the triangle. If one looks at the overall structure of the Euclidean definitions, one can recognize an u ...
... triangle for every triple of straight lines. Thus, the explicit Euclidean definition alone cannot be transformed into a postulate. We need to reveal a not explicitly mentioned characterization of the triangle. If one looks at the overall structure of the Euclidean definitions, one can recognize an u ...
Computational Geometry
... • What if only the vertices of the triangle are given? • Given 3 vertices (x1, y1), (x2, y2), (x3, y3) • Area = abs( x1*y2 + x2*y3 + x3*y1 - x2*y1 x3*y2 - x1*y3 ) / 2 • Note: abs can be omitted if the vertices are in counterclockwise order. If the vertices are in clockwise order, the difference eval ...
... • What if only the vertices of the triangle are given? • Given 3 vertices (x1, y1), (x2, y2), (x3, y3) • Area = abs( x1*y2 + x2*y3 + x3*y1 - x2*y1 x3*y2 - x1*y3 ) / 2 • Note: abs can be omitted if the vertices are in counterclockwise order. If the vertices are in clockwise order, the difference eval ...
Geometry
... G-SRT.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G-SRT.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G-SRT.2 Given two figures, use the definiti ...
... G-SRT.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G-SRT.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G-SRT.2 Given two figures, use the definiti ...
Discovery of Non-Euclidean Geometry
... with two base right angles. Analogously, ∠P DD0 is acute and ∠DCC 0 is obtuse. Of course ∠P DD0 < ∠DCC 0 . Then CC 0 < DD0 by property of quadrilaterals with two base right angles. We claim |AA0 | ≤ |P Q| for all A on open ray r̊(P, X). Suppose |AA0 | > |P Q|. Let S be a point on AA0 such that |AS| ...
... with two base right angles. Analogously, ∠P DD0 is acute and ∠DCC 0 is obtuse. Of course ∠P DD0 < ∠DCC 0 . Then CC 0 < DD0 by property of quadrilaterals with two base right angles. We claim |AA0 | ≤ |P Q| for all A on open ray r̊(P, X). Suppose |AA0 | > |P Q|. Let S be a point on AA0 such that |AS| ...